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Chapter 1: Problem 7

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=\frac{3}{x-4} \text { and } g(x)=\frac{3}{x}+4$$

Short Answer

Expert verified
The two functions \(f(x)=\frac{3}{x-4}\) and \(g(x)=\frac{3}{x}+4\) are inverse functions of each other.

Step by step solution

01

Find \(f(g(x))\)

Substitute \(g(x)\) into \(f(x)\), i.e. \(f(g(x))=f\left(\frac{3}{x}+4\right)=\frac{3}{\left(\frac{3}{x}+4\right)-4}\). Simplifying this results in \(x\).
02

Find \(g(f(x))\)

Now substitute \(f(x)\) into \(g(x)\), i.e. \(g(f(x))=g\left(\frac{3}{x-4}\right)=\frac{3}{\left(\frac{3}{x-4}\right)}+4\). Simplifying this we also get \(x\).
03

Determine if the functions are inverses

Since we found that both \(f(g(x))\) and \(g(f(x))\) are equal to \(x\), these two functions are indeed inverses of each other.

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